Abstract

A general method of surface profiling with phase-shifting interferometry techniques uses iterative linear regression to fit the sequence of interferograms to a physical model of the cavity. The physical model incorporates all important cavity influences, including environmentally induced rigid-body motion, phase shifter miscalibrations, multiple interference, geometry-induced spatial phase-shift variations, and their cross-couplings. By incorporating an initial estimate of the surface profile and iteratively solving for space- and time-dependent variables separately, convergence is robust and rapid. The technique has no restriction on surface shape or departure.

© 2014 Optical Society of America

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References

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  1. H. Schreiber and J. Bruning, “Optical shop testing,” in Phase Shifting Interferometry, D. Malacara, ed., 3rd ed. (Wiley, 2007), Chap. 14.
  2. P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  3. P. de Groot, “Correlated errors in phase shifting laser Fizeau interferometry,” Appl. Opt. 53, 4334–4342 (2014).
    [CrossRef]
  4. The techniques described in this paper are protected by U.S. patents7,796,273, 7,796,275, 7,948,639, and foreign patents or patents pending.
  5. L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,948,639 (24May2011).
  6. The technique described in this paper is marketed by Zygo Corporation under the name QPSI.
  7. I. Kong and S. Kim, “General algorithm for phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
    [CrossRef]
  8. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
    [CrossRef]
  9. G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. L. Deck, “Suppressing phase errors from vibration in phase shifting interferometry,” Appl. Opt. 48, 3948–3960 (2009).
    [CrossRef]
  14. P. de Groot, “Phase shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34, 2856–2863 (1995).
    [CrossRef]
  15. P. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef]
  16. L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,796,273 (14September2010).
  17. L. Deck, “Phase shifting interferometry in the presence of vibration using phase bias,” U.S. patent7,796,275 (14September2010).
  18. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [CrossRef]
  19. D. M. Sykora and M. L. Holmes, “Dynamic measurements using a Fizeau interferometer,” Proc. SPIE 8082, 80821R (2011).
    [CrossRef]

2014 (1)

2013 (1)

2011 (1)

D. M. Sykora and M. L. Holmes, “Dynamic measurements using a Fizeau interferometer,” Proc. SPIE 8082, 80821R (2011).
[CrossRef]

2009 (1)

2004 (1)

2000 (2)

1995 (3)

1994 (1)

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

1990 (1)

Bruning, J.

H. Schreiber and J. Bruning, “Optical shop testing,” in Phase Shifting Interferometry, D. Malacara, ed., 3rd ed. (Wiley, 2007), Chap. 14.

Chen, M.

de Groot, P.

Deck, L.

L. Deck, “Suppressing phase errors from vibration in phase shifting interferometry,” Appl. Opt. 48, 3948–3960 (2009).
[CrossRef]

L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,796,273 (14September2010).

L. Deck, “Phase shifting interferometry in the presence of vibration using phase bias,” U.S. patent7,796,275 (14September2010).

L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,948,639 (24May2011).

Guo, H.

Han, B.

Han, G. S.

He, J.

Holmes, M. L.

D. M. Sykora and M. L. Holmes, “Dynamic measurements using a Fizeau interferometer,” Proc. SPIE 8082, 80821R (2011).
[CrossRef]

Ji, F.

Kim, S.

I. Kong and S. Kim, “General algorithm for phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

Kim, S. W.

Kong, I.

I. Kong and S. Kim, “General algorithm for phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

Liu, Q.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Schreiber, H.

H. Schreiber and J. Bruning, “Optical shop testing,” in Phase Shifting Interferometry, D. Malacara, ed., 3rd ed. (Wiley, 2007), Chap. 14.

Sykora, D. M.

D. M. Sykora and M. L. Holmes, “Dynamic measurements using a Fizeau interferometer,” Proc. SPIE 8082, 80821R (2011).
[CrossRef]

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Wang, Y.

Wang, Z.

Wei, C.

Wizinowich, P.

Appl. Opt. (7)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Opt. Eng. (1)

I. Kong and S. Kim, “General algorithm for phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

D. M. Sykora and M. L. Holmes, “Dynamic measurements using a Fizeau interferometer,” Proc. SPIE 8082, 80821R (2011).
[CrossRef]

Other (6)

L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,796,273 (14September2010).

L. Deck, “Phase shifting interferometry in the presence of vibration using phase bias,” U.S. patent7,796,275 (14September2010).

H. Schreiber and J. Bruning, “Optical shop testing,” in Phase Shifting Interferometry, D. Malacara, ed., 3rd ed. (Wiley, 2007), Chap. 14.

The techniques described in this paper are protected by U.S. patents7,796,273, 7,796,275, 7,948,639, and foreign patents or patents pending.

L. Deck, “Phase shifting interferometry in the presence of vibration,” U.S. patent7,948,639 (24May2011).

The technique described in this paper is marketed by Zygo Corporation under the name QPSI.

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Figures (9)

Fig. 1.
Fig. 1.

Typical laser Fizeau interferometer system configured to measure a sphere.

Fig. 2.
Fig. 2.

MPSI process flow.

Fig. 3.
Fig. 3.

Typical plot of the surface RMS error and the mean merit function change as a function of the number of iterations. A merit function change threshold of 0.1% is typically used as an iteration termination condition. The arrows indicate which axis to use for which curve.

Fig. 4.
Fig. 4.

Maximum allowed pure piston vibration amplitude for convergence as a function of vibration frequency normalized to the frame rate when using the 13-frame algorithm.

Fig. 5.
Fig. 5.

Surface error maps for conventional PSI (left column) and MSPI (right column) from simulations of a 4%–40% fast spherical cavity with no vibrations (top), 20 nm amplitude vibrations (middle), and 60 nm amplitude vibrations (bottom). Note the different height scales.

Fig. 6.
Fig. 6.

Top profile represents a conventional PSI measurement of a vibrated flat cavity with about five fringes of tilt. The bottom profile shows the same data set processed with MPSI.

Fig. 7.
Fig. 7.

Top profile represents a conventional PSI measurement of a vibrated spherical cavity with four fringes of departure. The bottom profile shows the same data set processed with MPSI.

Fig. 8.
Fig. 8.

Surface profiles from a 4%–4% cavity measured with an F/0.65 transmission sphere undergoing large tilt and piston motions. The top profile is the surface using conventional PSI, exhibiting ripple and data loss from phase breakup. The bottom profile shows the same surface after MPSI processing.

Fig. 9.
Fig. 9.

Graphs of differences between PSI and MPSI surfaces from a 4%–40% fast (F/0.65) spherical cavity under quiet conditions for simulated (left) and real (right) data used to highlight the cross-coupling error. The color scale is set to ±5nm in both graphs.

Equations (14)

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I(x,t)=|rref+rtestexp[iΘ(x,t)]1+rrefrtestexp[iΘ(x,t)]|2,
I(x,t)=A(x)+V(x)k=1K(g)k1cos[kΘ(x,t)],
Θ(x,t)=Φ(x)+Δ(x,t),
Δ(x,t)=φ(t0)+α(t)x+β(t)y+[φ(t)φ(t0)]1ρ(x)2c2,
A(x)={max[Im(x,t)]+min[Im(x,t)]}/2V(x)={max[Im(x,t)]min[Im(x,t)]}/2,
φn(t)=φ(t)+φ(t)αn(t)=α(t)+α(t)βn(t)=β(t)+β(t)cn=c+c,
In(x,t)=I(x,t)++[φ(t)γ(x)+α(t)x+β(t)y+cη(x,t)]H(x,t),
H(x,t)=V(x)k=1K(g)k1ksin[kΘ(x,t)],
γ(x)=1ρ(x)2c2η(x,t)=[φ(t)φ(t0)]ρ(x)2c/γ(x).
χ(t)=12Mi=1M[Im(xi,t)In(xi,t)V(xi)]2,
I(x,t)=A(x)+k=1KCk(x)cos[kΔ(x,t)]+k=1KSk(x)sin[kΔ(x,t)],
Ck(x)=V(x)(g)k1cos[kΦ(x)]Sk(x)=V(x)(g)k1sin[kΦ(x)].
V(x)=C1(x)2+S1(x)2Φ(x)=tan1[S1(x)/C1(x)],
g=i=1MC2(xi)2+S2(xi)2i=1MV(xi).

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